# Analyzing Concavity of a Function

Inflection Points and Concavity Intuition

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## Key Questions

• In short, concavity is the rate of change of the slope of a function. Thus, it is the slope of the slope. In other words, it is the derivative of the derivative, or the second derivative.

There are three types of concavity: positive, negative, and 0. A function could have one type of concavity for its entire domain, or multiple types of concavity. Let's look at all three types of concavity.

1. Positive Concavity Let's assume we have a function that has a positive concavity for its entire domain. The rate of change of the slope is positive. In other words, the slope is always increasing. Examples are $y = {x}^{2} , y = 3 {x}^{2}$. Let's take $f \left(x\right) = 3 {x}^{2}$.

The first derivative of the function: $f ' \left(x\right) = 6 x$
The second derivative of the function: $f ' ' \left(x\right) = 6$
The concavity is positive since $6 > 0$. Graphically, functions that have positive concavity look like a capital 'U', or a smiley face, as shown below:

2. Negative Concavity Let's assume we have a function that has a negative concavity for its entire domain. The rate of change of the slope is negative. In other words, the slope is always decreasing. Examples are $y = - {x}^{2} , y = - 3 {x}^{2}$. Let's take $f \left(x\right) = - 3 {x}^{2}$.

The first derivative of the function: $f ' \left(x\right) = - 6 x$
The second derivative of the function: $f ' ' \left(x\right) = - 6$
The concavity is negative since $- 6 < 0$. Graphically, functions that have positive concavity look like an upside down 'U', or a hill, as shown below:

3. Concavity of 0 Let's look at functions that have multiple concavities. Let's take $f \left(x\right) = \sin \left(x\right)$.

The first derivative of the function: $f ' \left(x\right) = \cos \left(x\right)$
The second derivative of the function: $f ' ' \left(x\right) = - \sin \left(x\right)$

Whether $f ' ' \left(x\right) = - \sin \left(x\right)$ is positive, negative, or 0 depends on the value of $x$. Take a look at the graph of $f \left(x\right) = \sin \left(x\right)$:

Let's evaluate the second derivative at different points along the x-axis.

At x = $\frac{\pi}{2} , - \sin \left(\frac{\pi}{2}\right) = - 1$ This is negative concavity.
At x = $- \frac{\pi}{2} , - \sin \left(- \frac{\pi}{2}\right) = 1$ This is positive concavity.
At x = $0 , - \sin \left(0\right) = 0$. This is concavity of 0.

For the function $\sin \left(x\right)$, the point $x = 0$ is called an inflection point. And inflection point is a point at which the second derivative equals 0, AND where the second derivative changes sign (positive to negative or negative to positive).

Additionally, a function could have points at which the concavity does not exist. This means, at these point the function does not have a second derivative. To learn more about this, read more on inflection points and non-differentiable functions.

• You find the concavity of a function by finding where the second derivative is positive (concave up) and where it is negative (concave down).

For example, suppose we wanted to find where the graph of

$f \left(x\right) = {x}^{3} - \frac{3}{2} {x}^{2}$

is concave up and concave down. First we find the second derivative of f(x).

$f ' \left(x\right) = 3 {x}^{2} - 3 x$

and

$f ' ' \left(x\right) = 6 x - 3$

Now, we find any values where the second derivative is equal to zero.

$f ' ' \left(x\right) = 6 x - 3 = 0$

$6 x = 3 \Rightarrow x = \frac{1}{2}$

Next we use test points to the right and left of x = $\frac{1}{2}$ to set up a sign chart so we can find intervals where the second derivative is positive or negative.

So, the graph of f(x) is concave down on (-oo, 1/2) and concave up on $\left(\frac{1}{2} , \infty\right)$. See the graph below.

The point where the concavity changes is called the point of inflection.

• For a quadratic function $f \left(x\right) = a {x}^{2} + b x + c$,
if $a > 0$, then $f$ is concave upward everywhere,
if $a < 0$, then $f$ is concave downward everywhere.

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